Solve for $x$ in the equation
\[2^{(16^x)} = 16^{(2^x)}.\]
Answer: We can write
\[16^{(2^x)} = (2^4)^{(2^x)} = 2^{4 \cdot 2^x}.\]Then $2^{16^x} = 2^{4 \cdot 2^x},$ so
\[16^x = 4 \cdot 2^x.\]In turn, we can write this as
\[2^{4x} = 2^{x + 2},\]so $4x = x + 2.$  Therefore, $x = \boxed{\frac{2}{3}}.$